Introduction to Graph Theory (2 page)

In Euclidean geometry the
objects
are a plane, some points, and some lines. The plane corresponds to a chessboard, the points and lines to chessmen. The
opening arrangement
is the list of axioms, which are accepted without proof. The analogy with the opening arrangement of chessmen may not be apparent, but it is quite strong. First, the opening arrangement of chessmen is
given;
to play chess you must start with that arrangement and no other. In the same way the axioms of Euclidean geometry are given. Second, the opening arrangement of chessmen specifies how the objects with which the game is played are related at the outset. This is exactly what the axioms do for the game of Euclidean geometry; they tell us, for example, that points and lines lie in the plane, that through two points there
passes one and only one line, etc. The
rules
of Euclidean geometry are the rules of formal logic, which is nothing but an etherealized version of the “common sense” we absorb from the culture as we grow up. Its rules tell us how statements can be combined to produce other statements. They tell us, for example, that the statements “All men are mortal” and “Plato is a man” yield the statement “Plato is mortal.” (The example is Aristotle's. He wrote the first book on logic by recording patterns of inference he saw people using every day.) In particular the rules of logic tell us how to create, from the opening arrangement (the list of axioms), new arrangements (called “theorems”). And the
goal
of Euclidean geometry is to produce as many “nice” arrangements as possible, that is to prove profound and surprising theorems. Checkmate terminates a chessmatch, but Euclidean geometry is open-ended.

Games have one more feature in common with pure mathematics. It is subtle but important. It is that the objects with which a game is played have no meaning outside the context of the game.

Chessmen, for example, are significant only in reference to chess. They have no necessary correspondence with anything external to the game. Of course, we could
interpret
the pieces as regiments at the First Battle of the Marne, and the board as French countryside. Or an interpretation could be brought about by a wager, say each piece represents $5.00 and losing it means paying that amount. But no such correspondence between the game and things outside the game is necessary.

You may balk at this, since, for historical reasons, chessmen have names and shapes that imply an essential correspondence with the external world. The key word is “essential”; there is indeed a correspondence, but it is inessential. After all, chessmen require names of some sort, and must be shaped differently to avoid confusion, so why not call them “kings”, “queens”, “bishops”, “knights”, etc., shape them accordingly, and trade on the image of excitement and competition thereby created? Doing so is harmless and makes the game more popular. But this particular interpretation of the game, like all others, has nothing to do with chess
per se.

Here's an example. Suppose we substitute silver dollars for kings, half-dollars for queens, quarters for bishops, dimes for knights, nickels for rooks, pennies for pawns, and an eight-by-eight array of chartreuse and violet circles for the standard board, but otherwise follow the rules of chess. Such a game would look strange and even sound strange—“pawn to king four” would now be “penny to silver dollar four”—but surely if we were to play this apparently unfamiliar game, there
would be no doubt that we are playing the familiar game of chess. Indeed, chessmasters sometimes play without a board or pieces of any kind; they merely announce the moves and keep track in their heads. Two such people are still playing chess, for after all they say they are, and they certainly should know.

It appears then that the essence of chess is its abstract structure. Names and shapes of pieces, colors of squares, whether the “squares” are in fact square, even the physical existence of board and pieces, are all irrelevant. What is relevant is the number and geometric arrangement of the “squares”, the number of types of piece and the number of pieces of each type, the quantitative-geometric power of each piece, etc. Everything else is a visual aid or a fairy tale.

So it is with pure mathematics. Euclid's words “plane”, “point”, and “line” suggest that geometry deals with flat surfaces, tiny dots, and stretched strings, but this implied interpretation of geometry is only that. It is analogous to the interpretation of chess as a battle. Geometry is no more a study of flat surfaces and dots than chess is a military exercise. As in any game, the objects geometers play with, and consequently their arrangements—the axioms and theorems—have no necessary correspondence with things external to the game.

In support of this let me point out that geometers never define the words “plane”, “point”, or “line”. (Euclid offered an intuitive explanation but did not actually define them; moderns leave the words undefined.) So
no one knows
what planes, points, or lines are, except to say that they are objects which are related to one another in accordance with the axioms. The three words are merely convenient names for the three types of object geometers play with. Any other names would do as well. Were we to attack Euclid's
Elements
with an eraser and remove every occurrence of the words “plane”, “point”, and “line”, replacing them respectively with the symbols “#”, “$”, and “?”, the result would still be
The Elements
and the game would still be geometry. To a casual observer the vandalized
Elements
wouldn't look like geometry; what had been “two points determine one and only one line” would now be “two $'s determine one and only one ?”. But then two people hunched over a board of chartreuse and violet circles, littered with coins, doesn't look like a chessmatch. The game would still be geometry because it would be structurally identical to geometry. And were we to further maim
The Elements
by erasing all the diagrams, it still wouldn't make a difference. Geometric diagrams are to geometers what board and pieces are to chessmasters: visual aids, helpful but not indispensable.

Why study pure mathematics?

There emerges from the foregoing an image of pure mathematics as a meaningless intellectual pastime. Yet carved over the door to Plato's Academy was the admonition, “Let no one ignorant of geometry enter here!” And pure mathematics has been held in the highest regard ever since. It would seem to have no more to recommend its inclusion in school curricula than, for instance, chess, yet it is universally favored by academics over other games. I shall give three reasons for this.

Pure mathematics is applicable.
Because pure mathematics has no inherent correspondence with the outside world, we are free to make it correspond, to interpret it, in any way we choose. And it so happens—this is the interesting part—that most branches of pure mathematics can be interpreted in such a way that the axioms and theorems become approximately true statements about the external world. In fact, some branches have several such interpretations.

Pure mathematics that has been made to correspond in this way to the world outside is called “applied mathematics”. Pure mathematics is Euclid saying “three $'s not on the same ? determine a unique #.” Applied mathematics is a surveyor reading Euclid, interpreting and “#” in a way that seems in accord with the axioms, and concluding that a tripod would be the most stable support for his telescope.

On one level, the applicability of pure mathematics is no surprise. Just as chess (as we know it) has been modeled on certain aspects of medieval warfare, even though strictly speaking the game has nothing to do with warfare, so too most branches of pure mathematics have started as models of physical situations. A branch of pure mathematics utterly lacking in significant interpretations would be boring to the community of pure mathematicians and would soon die out from lack of interest. Though they are unconcerned with applications as such, pure mathematicians are like most people in that they find it hard to be enthusiastic about something unless, under some aspect at least, it has the spontaneous appearance of truth.

But on a deeper level, the applicability of pure mathematics is quite mysterious. It's true that pure mathematics often originates with an abstraction from the physical world, as geometry begins with idealized dots and strings and tabletops, but the tie is only historical. Once the abstractions have been made the mathematical game comes into independent existence and evolves under its own laws. It has no necessary correspondence with the original physical situation. The
mathematician does not deal with physical objects themselves, but with idealizations that exist independently and differ from their physical counterparts in a great many respects. And entirely within his own mind, the mathematician subjects these abstractions to a reflective, self-analytic process, a process in which he is trying to learn about himself, to learn what in a sense he already knows. This process is strictly internal to a human mind—a Western mind at that—and so is presumably different from whatever the process by which the physical situation evolves; yet when the mathematician compares his results to outside events, he often finds that nature has evolved to a state remarkably like his mathematical model. That the universe is so constructed has seemed uncanny to many famous mathematicians and scientists, moving them to comment in a mystical fashion that seems totally out of character:

“Number rules the universe.”

—Pythagoras

“Mathematics is the only true metaphysics.”

—Lord Kelvin

“How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?”

—Einstein

“The Great Architect of the Universe now begins to appear as a pure mathematician. ”

—Sir James Jeans

(Quotations from E. T. Bell's
Men of Mathematics.)

Applicability is the chief difference between the games known collectively as “pure mathematics” and other games. There's some kind of chemistry involving nature, and people, and pure mathematics, that enables applied mathematics to predict the future, whereas mankind has yet to make a success of applied chess.

Pure mathematics is a culture clue
.

“. . . common sense is, as a matter of fact, nothing more than layers of preconceived notions stored in our memories and emotions for the most part before age eighteen.”

—Albert Einstein

Our common sense, or world view, is not “common” to all people. It is shaped by the culture we inhabit. It is like a pair of glasses few of us ever manage to take off, so of course we see confirmation everywhere we look.

Much of Western intellectual tradition has been inherited from the Greeks. Our science and philosophy in particular are shot through with beliefs and opinions and forms of speech that were once explicit doctrines of Plato, Aristotle, and the like, but have come to be embedded anonymously in the fabric of our thought. Of this embedded material perhaps the most fundamental is logic, the standard by which we judge reasoning to be “correct”, a standard first written down by Aristotle in
The Organon
(about 350 B.C.).

Is logic itself “correct”? Some Eastern philosophers would call it “ignorance”. I use logic all the time in mathematics, and it seems to yield “correct” results, but in mathematics “correct” by and large means “logical”, so I'm back where I started. I can't defend logic because I can't remove my glasses.

“Correct” or not, logic is basic to Western rationality and to the whole scientific enterprise. And not surprisingly, since logic is the study of deduction and pure mathematics is the only completely deductive study, logic is inextricably intertwined with pure mathematics. I think this is the chief reason for the prominence of mathematics in our schools. Logic is a fundamental component of the culture, so the culture quite naturally sets a premium on teaching the next generation to think in logical categories.

Incidentally, there's a lot of debate on which came first, logic or mathematics. In one sense logic is prior to mathematics, as mathematics uses the laws of logic. But Aristotle abstracted the laws of logic at least in part from the pure mathematics he studied at Plato's Academy, so in another sense mathematics is more basic. G. Spencer Brown argues this position in
Laws of Form,
p. 102:

A theorem is no more proved by logic and computation than a sonnet is written by grammar and rhetoric, or than a sonata is composed by harmony and counterpoint, or a picture painted by balance and perspective. Logic and computation, grammar and rhetoric, harmony and counterpoint, balance and perspective, can be seen in the work
after
it is created, but these forms are, in the final analysis, parasitic on, they have no existence apart from, the creativity of the work itself. Thus the relation of logic to mathematics is seen to be that of an applied science to its pure ground, and all applied science is seen as drawing sustenance from a process of creation with which it can combine to give structure, but which it cannot appropriate.

Pure mathematics is fun.
At this moment there are thousands of people around the world doing pure mathematics. A few might be doing so because they foresee a possible application. A few might be philosophers taking Bertrand Russell's advice that “to create a healthy philosophy you should renounce metaphysics but be a good mathematician.” There might even be a few ascetics who are doing it to sharpen their minds. But the vast majority are doing it simply because it's fun.

Pure mathematics is a first-rate intellectual adventure, “. . . an independent world/Created out of pure intelligence” (Wordsworth) that is neither science nor art but somehow partakes of both.

Pure mathematics is the world's best game. It is more absorbing than chess, more of a gamble than poker, and lasts longer than Monopoly. It's free. It can be played anywhere—Archimedes did it in a bathtub. It is dramatic, challenging, endless, and full of surprises.